In [1]} the authors characterized the full range of \((p,q)\)  on which the Carleman inequality (REF ) holds. More precisely, under the assumption \(1<p,q<\infty \) , it was proved that (REF ) holds if and only if
\( \tfrac{1}{p}-\tfrac{1}{q}=\tfrac{2}{d} \ \ \text{and} \ \ \tfrac{d^2-4}{2d(d-1)}\le \tfrac{1}{p} \le \tfrac{d+2}{2(d-1)}. \) 
